A $\frac{4}{3}$-approximation algorithm for half-integral cycle cut instances of the TSP

Abstract

A long-standing conjecture for the traveling salesman problem (TSP) states that the integrality gap of the standard linear programming relaxation of the TSP (sometimes called the Subtour LP or the Held-Karp bound) is at most 4/3 for symmetric instances of the TSP obeying the triangle inequality; that is, the cost of an optimal tour is at most 4/3 times the value of the value of the corresponding linear program. There is a variety of evidence in support of the conjecture (see, for instance, Goemans in Math Program 69:335–349, 1995; Benoit and Boyd in Math Oper Res 33:921–931, 2008). It has long been known that the integrality gap is at most 3/2 (Wolsey in Math Program Study 13:121–134, 1980; Shmoys and Williamson in Inf Process Lett 35:281–285, 1990). Despite significant efforts by the community, the conjecture remains open. In this paper we consider the half-integral case, in which a feasible solution to the LP has solution values in $\{0, 1/2, 1\}$. Such instances have been conjectured to be the most difficult instances for the overall four-thirds conjecture (Schalekamp et al. in Math Oper Res 39(2):403–417, 2014). Karlin et al. (in: Proceedings of the 52nd Annual ACM Symposium on the the Theory of Computing, ACM, New York, 2020), in a breakthrough result, were able to show that in the half-integral case, the integrality gap is at most 1.49993; Gupta et al. (in: Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, 2022. https://arxiv.org/abs/2111.09290) showed a slight improvement of this result to 1.4983. Additionally, this result led to the first significant progress on the overall conjecture in decades; the same authors showed the integrality gap of the Subtour LP is at most $1.5-\epsilon$ for some $\epsilon >10^{-36}$ Karlin et al. in 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS). https://doi.org/10.1109/FOCS54457.2022.00084. With the improvements on the 3/2 bound remaining very incremental, even in the half-integral case, we turn the question around and look for a large class of half-integral instances for which we can prove that the 4/3 conjecture is correct, preferably one containing the known worst-case instances. In Karlin et al.’s work on the half-integral case, they perform induction on a hierarchy of critical tight sets in the support graph of the LP solution, in which some of the sets correspond to cycle cuts and the others to degree cuts. Here we show that if all the sets in the hierarchy correspond to cycle cuts, then we can find a distribution of tours whose expected cost is at most 4/3 times the value of the half-integral LP solution; sampling from the distribution gives us a randomized 4/3-approximation algorithm. We note that two important bad cases with an integrality gap of 4/3 have a half-integral LP solution in which all the critical tight sets in the hierarchy are cycle cuts; thus our result is tight. Our overall approach is novel. Most recent work has focused on showing that some variation of the Christofides-Serdyukov algorithm (Christofides in “Worst case analysis of a new heuristic for the traveling salesman problem”. Report 388, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, 1976; Serdyukov in Upravlyaemye Sistemy 17:76–79, 1978) that combines a randomly sampled spanning tree plus a T-join (or a matching) can be shown to give a bound better than 1.5. Here we show that for any point in a region of “patterns” of edges incident to each cycle cut, we can give a distribution of patterns connecting all the child cycle cuts such that the distribution of patterns for each child also falls in the region. This region gives rise to a distribution on Eulerian tours in which each edge in the support of the LP is used at most four-thirds of its LP value of the time, which then gives the result.